# Properties

 Label 980.1.ba.b Level $980$ Weight $1$ Character orbit 980.ba Analytic conductor $0.489$ Analytic rank $0$ Dimension $6$ Projective image $D_{7}$ CM discriminant -20 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$980 = 2^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 980.ba (of order $$14$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.489083712380$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{14})$$ Defining polynomial: $$x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{7}$$ Projective field: Galois closure of 7.1.110730297608000.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{14} q^{2} + ( -1 - \zeta_{14}^{6} ) q^{3} + \zeta_{14}^{2} q^{4} -\zeta_{14}^{3} q^{5} + ( 1 - \zeta_{14} ) q^{6} -\zeta_{14}^{2} q^{7} + \zeta_{14}^{3} q^{8} + ( 1 - \zeta_{14}^{5} + \zeta_{14}^{6} ) q^{9} +O(q^{10})$$ $$q + \zeta_{14} q^{2} + ( -1 - \zeta_{14}^{6} ) q^{3} + \zeta_{14}^{2} q^{4} -\zeta_{14}^{3} q^{5} + ( 1 - \zeta_{14} ) q^{6} -\zeta_{14}^{2} q^{7} + \zeta_{14}^{3} q^{8} + ( 1 - \zeta_{14}^{5} + \zeta_{14}^{6} ) q^{9} -\zeta_{14}^{4} q^{10} + ( \zeta_{14} - \zeta_{14}^{2} ) q^{12} -\zeta_{14}^{3} q^{14} + ( -\zeta_{14}^{2} + \zeta_{14}^{3} ) q^{15} + \zeta_{14}^{4} q^{16} + ( -1 + \zeta_{14} - \zeta_{14}^{6} ) q^{18} -\zeta_{14}^{5} q^{20} + ( -\zeta_{14} + \zeta_{14}^{2} ) q^{21} + ( \zeta_{14}^{5} - \zeta_{14}^{6} ) q^{23} + ( \zeta_{14}^{2} - \zeta_{14}^{3} ) q^{24} + \zeta_{14}^{6} q^{25} + ( -1 - \zeta_{14}^{4} + \zeta_{14}^{5} - \zeta_{14}^{6} ) q^{27} -\zeta_{14}^{4} q^{28} + ( 1 - \zeta_{14}^{3} ) q^{29} + ( -\zeta_{14}^{3} + \zeta_{14}^{4} ) q^{30} + \zeta_{14}^{5} q^{32} + \zeta_{14}^{5} q^{35} + ( 1 - \zeta_{14} + \zeta_{14}^{2} ) q^{36} -\zeta_{14}^{6} q^{40} + ( \zeta_{14}^{2} + \zeta_{14}^{4} ) q^{41} + ( -\zeta_{14}^{2} + \zeta_{14}^{3} ) q^{42} + ( \zeta_{14}^{3} + \zeta_{14}^{5} ) q^{43} + ( -\zeta_{14} + \zeta_{14}^{2} - \zeta_{14}^{3} ) q^{45} + ( 1 + \zeta_{14}^{6} ) q^{46} + ( -\zeta_{14}^{4} + \zeta_{14}^{5} ) q^{47} + ( \zeta_{14}^{3} - \zeta_{14}^{4} ) q^{48} + \zeta_{14}^{4} q^{49} - q^{50} + ( 1 - \zeta_{14} - \zeta_{14}^{5} + \zeta_{14}^{6} ) q^{54} -\zeta_{14}^{5} q^{56} + ( \zeta_{14} - \zeta_{14}^{4} ) q^{58} + ( -\zeta_{14}^{4} + \zeta_{14}^{5} ) q^{60} + ( -\zeta_{14} + \zeta_{14}^{2} ) q^{61} + ( -1 + \zeta_{14} - \zeta_{14}^{2} ) q^{63} + \zeta_{14}^{6} q^{64} -2 q^{67} + ( \zeta_{14}^{4} - 2 \zeta_{14}^{5} + \zeta_{14}^{6} ) q^{69} + \zeta_{14}^{6} q^{70} + ( \zeta_{14} - \zeta_{14}^{2} + \zeta_{14}^{3} ) q^{72} + ( \zeta_{14}^{5} - \zeta_{14}^{6} ) q^{75} + q^{80} + ( 1 - \zeta_{14}^{3} + \zeta_{14}^{4} - \zeta_{14}^{5} + \zeta_{14}^{6} ) q^{81} + ( \zeta_{14}^{3} + \zeta_{14}^{5} ) q^{82} + ( \zeta_{14} - \zeta_{14}^{4} ) q^{83} + ( -\zeta_{14}^{3} + \zeta_{14}^{4} ) q^{84} + ( \zeta_{14}^{4} + \zeta_{14}^{6} ) q^{86} + ( -1 - \zeta_{14}^{2} + \zeta_{14}^{3} - \zeta_{14}^{6} ) q^{87} + ( -\zeta_{14} + \zeta_{14}^{4} ) q^{89} + ( -\zeta_{14}^{2} + \zeta_{14}^{3} - \zeta_{14}^{4} ) q^{90} + ( -1 + \zeta_{14} ) q^{92} + ( -\zeta_{14}^{5} + \zeta_{14}^{6} ) q^{94} + ( \zeta_{14}^{4} - \zeta_{14}^{5} ) q^{96} + \zeta_{14}^{5} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + q^{2} - 5q^{3} - q^{4} - q^{5} + 5q^{6} + q^{7} + q^{8} + 4q^{9} + O(q^{10})$$ $$6q + q^{2} - 5q^{3} - q^{4} - q^{5} + 5q^{6} + q^{7} + q^{8} + 4q^{9} + q^{10} + 2q^{12} - q^{14} + 2q^{15} - q^{16} - 4q^{18} - q^{20} - 2q^{21} + 2q^{23} - 2q^{24} - q^{25} - 3q^{27} + q^{28} + 5q^{29} - 2q^{30} + q^{32} + q^{35} + 4q^{36} + q^{40} - 2q^{41} + 2q^{42} + 2q^{43} - 3q^{45} + 5q^{46} + 2q^{47} + 2q^{48} - q^{49} - 6q^{50} + 3q^{54} - q^{56} + 2q^{58} + 2q^{60} - 2q^{61} - 4q^{63} - q^{64} - 12q^{67} - 4q^{69} - q^{70} + 3q^{72} + 2q^{75} + 6q^{80} + 2q^{81} + 2q^{82} + 2q^{83} - 2q^{84} - 2q^{86} - 3q^{87} - 2q^{89} + 3q^{90} - 5q^{92} - 2q^{94} - 2q^{96} + q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$197$$ $$491$$ $$\chi(n)$$ $$\zeta_{14}^{6}$$ $$-1$$ $$-1$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
239.1
 −0.623490 + 0.781831i 0.222521 + 0.974928i 0.900969 + 0.433884i 0.900969 − 0.433884i 0.222521 − 0.974928i −0.623490 − 0.781831i
−0.623490 + 0.781831i −1.62349 0.781831i −0.222521 0.974928i −0.900969 0.433884i 1.62349 0.781831i 0.222521 + 0.974928i 0.900969 + 0.433884i 1.40097 + 1.75676i 0.900969 0.433884i
379.1 0.222521 + 0.974928i −0.777479 0.974928i −0.900969 + 0.433884i 0.623490 + 0.781831i 0.777479 0.974928i 0.900969 0.433884i −0.623490 0.781831i −0.123490 + 0.541044i −0.623490 + 0.781831i
519.1 0.900969 + 0.433884i −0.0990311 0.433884i 0.623490 + 0.781831i −0.222521 0.974928i 0.0990311 0.433884i −0.623490 0.781831i 0.222521 + 0.974928i 0.722521 0.347948i 0.222521 0.974928i
659.1 0.900969 0.433884i −0.0990311 + 0.433884i 0.623490 0.781831i −0.222521 + 0.974928i 0.0990311 + 0.433884i −0.623490 + 0.781831i 0.222521 0.974928i 0.722521 + 0.347948i 0.222521 + 0.974928i
799.1 0.222521 0.974928i −0.777479 + 0.974928i −0.900969 0.433884i 0.623490 0.781831i 0.777479 + 0.974928i 0.900969 + 0.433884i −0.623490 + 0.781831i −0.123490 0.541044i −0.623490 0.781831i
939.1 −0.623490 0.781831i −1.62349 + 0.781831i −0.222521 + 0.974928i −0.900969 + 0.433884i 1.62349 + 0.781831i 0.222521 0.974928i 0.900969 0.433884i 1.40097 1.75676i 0.900969 + 0.433884i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 939.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
49.e even 7 1 inner
980.ba odd 14 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 980.1.ba.b yes 6
4.b odd 2 1 980.1.ba.a 6
5.b even 2 1 980.1.ba.a 6
20.d odd 2 1 CM 980.1.ba.b yes 6
49.e even 7 1 inner 980.1.ba.b yes 6
196.k odd 14 1 980.1.ba.a 6
245.p even 14 1 980.1.ba.a 6
980.ba odd 14 1 inner 980.1.ba.b yes 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
980.1.ba.a 6 4.b odd 2 1
980.1.ba.a 6 5.b even 2 1
980.1.ba.a 6 196.k odd 14 1
980.1.ba.a 6 245.p even 14 1
980.1.ba.b yes 6 1.a even 1 1 trivial
980.1.ba.b yes 6 20.d odd 2 1 CM
980.1.ba.b yes 6 49.e even 7 1 inner
980.1.ba.b yes 6 980.ba odd 14 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 5 T_{3}^{5} + 11 T_{3}^{4} + 13 T_{3}^{3} + 9 T_{3}^{2} + 3 T_{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(980, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6}$$
$3$ $$1 + 3 T + 9 T^{2} + 13 T^{3} + 11 T^{4} + 5 T^{5} + T^{6}$$
$5$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6}$$
$7$ $$1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6}$$
$11$ $$T^{6}$$
$13$ $$T^{6}$$
$17$ $$T^{6}$$
$19$ $$T^{6}$$
$23$ $$1 - 4 T + 9 T^{2} - 8 T^{3} + 4 T^{4} - 2 T^{5} + T^{6}$$
$29$ $$1 - 3 T + 9 T^{2} - 13 T^{3} + 11 T^{4} - 5 T^{5} + T^{6}$$
$31$ $$T^{6}$$
$37$ $$T^{6}$$
$41$ $$1 + 4 T + 9 T^{2} + 8 T^{3} + 4 T^{4} + 2 T^{5} + T^{6}$$
$43$ $$1 - 4 T + 9 T^{2} - 8 T^{3} + 4 T^{4} - 2 T^{5} + T^{6}$$
$47$ $$1 + 3 T + 2 T^{2} - T^{3} + 4 T^{4} - 2 T^{5} + T^{6}$$
$53$ $$T^{6}$$
$59$ $$T^{6}$$
$61$ $$1 + 4 T + 9 T^{2} + 8 T^{3} + 4 T^{4} + 2 T^{5} + T^{6}$$
$67$ $$( 2 + T )^{6}$$
$71$ $$T^{6}$$
$73$ $$T^{6}$$
$79$ $$T^{6}$$
$83$ $$1 - 4 T + 9 T^{2} - 8 T^{3} + 4 T^{4} - 2 T^{5} + T^{6}$$
$89$ $$1 + 4 T + 9 T^{2} + 8 T^{3} + 4 T^{4} + 2 T^{5} + T^{6}$$
$97$ $$T^{6}$$